Consider an envelope detector that is used to detect the message sent in the AM system shown in the examples. The envelope detector as a system is composed of two cascaded systems: one which computes the absolute value of the input (implemented with ideal diodes), and a second that low-pass filters its input (implemented with an RC circuit). The following is an implementation of these operations in the discrete time so we can use numeric MATLAB. Let the input to the envelope detector be

x(t) = [p(t) + P]cos(Ω₀t).

Use MATLAB to solve numerically this problem.

(a) Consider first p(t) = 20[u(t) − u(t − 40)] − 10[u(t − 40)−u(t − 60)] and let Ω₀ = 2π, P = 1.1|min(p(t)|. Generate the signals p(t) and x(t) for 0 ≤ t ≤ 100 with and interval of T_s = 0.01.

(b) Consider then the subsystem that computes the absolute value of the input x(t). Plot y(t) = ∣x(t)∣.

(c) Compute the low-pass filtered signal (h * y)(t) by using an RC circuit with impulse response h(t) = e^{−0.8 t} u(t). To implement the convolution use convfunction multiplied by T_s. Plot together the message signal p(t), the modulated signal x(t), the absolute value y(t), and the envelope z(t) = (h*y)(t) − P. Does this envelope look like p(t)?

(d) Consider the message signal p(t) = 2 cos(0.2π t), Ω₀ = 10 π and P =|min(p(t)|and repeat the process. Scale the signal to get the original p(t).